Magnetic anisotropies of Ho(III) and Dy(III) single-molecule magnets experimentally determined via polarized neutron diffraction

We present the magnetic anisotropy of two isostructural pentagonal-bipyramidal complexes, [Ln(H2O)5(HMPA)2]I3·2HMPA (HMPA = hexamethylphosphoramide, Ln = Dy, Ho). Using ac magnetic susceptibility measurements, we find magnetic relaxation barriers of 600 K and 270 K for the Dy- and Ho-compounds, respectively. This difference is supported by polarized neutron diffraction (PND) measured at 5 K and 1 T which provides the first experimental evidence that the transverse elements in the magnetic anisotropy of the Ho-analogue are significant, whereas the Dy-analogue has a near-axial magnetic anisotropy with vanishing transverse contributions. The coordination geometries of the two complexes are highly similar, and we attribute the loss of strong magnetic axiality as expressed in the atomic susceptibility tensors from PND, as well as the smaller relaxation barrier in the Ho-complex compared to the Dy-complex, to the less favorable interaction of the pentagonal bipyramidal crystal field with the characteristics of the Ho(III) 4f-charge distribution.


Crystallographic details
The structure of 2 was solved using ShelXT 1 and refined using ShelXL 2 within the program Olex2. 3 Positional parameters and anisotropic atomic displacement parameters (adps) were refined for all non-Hydrogen atoms, some of the atoms of the HMPA (Hexamethylphophoramide) were modelled over two partially occupied positions (0.7:0.3 for the coordinated HMPA and 0.5:0.5 for the free HMPA) as part of a disorder model, where isotropic adps were retained.Suitable distance and adp similarity restraints were applied to these atoms.Hydrogen atoms were placed in calculated positions and included as part of a riding model.The full details are available in the CIF (CCDC 2057601).2.321 (7)   O1-Ho1-O2 179.6 (4) O2-Ho1-O5W 90.9 (4) O1-Ho1-O1W 90.Table S3.Shape analysis for complexes 1 and 2. The lowest CShMs value, is highlighted.Continuous shape measures analysis estimates the distortion from the perfect polyhedron, where 0 corresponds to the ideal structure. 4

Magnetic Properties
Figure S3.Reduced magnetization plot for 2 at 2 K, 4 K and 6 K.

Ab initio calculations
Calculations on 2 were carried out as complete active space self-consistent field (CASSCF) calculations using the CASSCF/RASSI-SO/SINGLE_ANISO approach in OpenMolcas version 18.09 through the pymolcas interface. 8,9 he structure determined from X-ray diffraction was used for the molecular geometry of the compound, including solvent atoms and counterions.For the disordered parts of the crystal structure, the groups with the largest occupancy were used in the calculations.The basis sets employed for the atoms in the structure were of the ANO-RCC family, using 8s7p5d3f2g1h (pVTZ) for Ho, 7s6p4d2f1g (pVTZ) for I, 3s2p1d (pVDZ) for O, 4s3p (VDZ) for P, 3s2p (VDZ) for N and C and 2s (VDZ) for H.The calculations were performed with 35 quintuplets arising from 10 f-electrons in Ho(III).
Table S4.CASSCF+RASSI-SO computed relative energies (in K) along with g tensors and deviations from the principal magnetisation axis for complex 2.
a Only >10% contributions are given.

Using X-ray structures for refinement against polarized neutron diffraction data
For this study, we did not have access to unpolarized neutron diffraction data to refine the nuclear structures for 1 or 2. To investigate the credibility of instead using an structure refined from x-ray diffraction when refining site susceptibilities with PND data, we studied the effect of exchanging the structural model used to refine a site susceptibility tensor against the same PND data.It is well established that X-ray crystal structures lead to artificially short bond X-H bond lengths, because electron density is shifted from near the nucleus of the hydrogen atom into the covalent bond.
Therefore, three different ways of obtaining the structural model were studied: X-ray structure with as-refined bond lengths to hydrogen (Xshort), X-ray structure with bond lengths to H equal to those obtained from neutron diffraction refinement (Xlong), conventional neutron diffraction structure refinement (ND).The study was undertaken for two different molecules, Dy(dbm)3bpy (dbm=1,3diphenyl-1,3-propanedione) (3) and CoBr2(tmtu)2 (tmtu=1,1,3,3-tetramethylthiourea) (4).For 3, both the nuclear structure along with the refinement of the PND-data and the X-ray structure have earlier been reported, 10,11 and for 4 the X-ray structure had also been published. 12The complete refinement of the nuclear structure of 4 and the execution of the PND experiment will be reported elsewhere. 13Here, the focus is on the effect of exchanging the structural model in the PND refinement.
The results for 3 in terms of tensor elements (  ) are shown in Table S6 along with the total value of  2 for six orientations of the magnetic field with respect to the crystal of the compound that was being measured.In addition, we show the angular discrepancy between the easy-axis direction for the neutron diffraction refinement and the other two models.A final comparison was made between the three tensors, based on a similarity index originally developed for anisotropic displacement tensors, but used here for the magnetic susceptibility tensors. 14The index, S12, rates the percentage difference between two second rank tensors, such that a value of 0 is obtained for fully identical tensors.For 4, in addition to probing the two different between X-ray models (Xshort and Xlong), we used two different approaches to refining H-atom positions from the X-ray diffraction data in SHELXL. 2 In one approach, the H-positions were refined with the "AFIX 137"-keyword that restricts a CH3-group to have tetrahedral angles but allows it to rotate freely around the fourth bond to C. In the other approach, "AFIX 33" was used, which keeps an ideal tetrahedral geometry, but fixes the rotation angle such that the CH3-group becomes staggered.The results for four of the models are shown in Table S7.For Xlong with the "AFIX 33"-keyword a successful refinement against the diffraction data could not be completed.The refinement was performed against four directions of applied magnetic field with respect to the molecule in the same beamline setup as that described in the main text.In the results of 3, we see that the susceptibility tensors obtained with an X-ray structure are very similar to the tensor obtained based on a nuclear structure.The Xlong-model for 3, gives a slightly better  2 -value and a smaller angular discrepancy, while the Xshort-model gives a smaller, and thus better, similarity index.Overall, the difference between the models is negligible, and on the order of the uncertainty of the model.For 4, the agreement parameters for Xshort(33) show that this model is unviable for the description of the magnetic susceptibility tensor of 4. For the two other approaches based on X-ray diffraction, there is no clear distinction between the models.Xlong(137) gives slightly better agreement with the ND-model, based both  2 , easy axis angle and similarity index, but similar to the results for 3, the difference between the models is estimated to be on the order of the model uncertainty.These results show that it is viable to use an X-ray structure to refine susceptibility tensors based on PND and that the elongation of bonds to H-atoms may provide an improvement to the modelling of PND-data.
With these results in mind, for this study we used structures with elongated bonds to H-atoms.The only bonds to H that are in the structures of 1 and 2 are the C-H-bonds in the CH3-groups of the HMPA-ligands and the O-H-bonds in the equatorial water molecules.A survey of known molecules in the CSD found that from neutron diffraction studies the C-H-bonds have mean lengths of 1.077(29) Å 16 .The bond lengths in the methyl groups of 1 and 2 were set to these values, and CH3-groups were refined using "AFIX 137".The bond lengths between O and H in the equatorial water molecules were not changed, but kept at the values obtained from a refinement using DFIX restraint of 0.85 Å with an esd of 0.02 Å.In the structure solution of 2, the bond lengths in the equatorial water molecules had already been fixed to successfully refine the structure, and in both structures, the hydrogen positions in the equatorial water molecules are not only influenced by the bonding to O, but also by the interaction with I --counterions and the solvent HMPA-ligands.Therefore, these bond lengths were left as refined.The change in R-values for 1 and 2 on going from the as-refined bond lengths (Hshort) to the elongated bond lengths (Hlong) are small and presented in Table S8.We note also that the use of an X-ray structure determination for refinement against PND-data has earlier been reported. 17

PND data reduction and refinement details
The data collection on 1 and 2 was performed as described in the main text.To keep the crystal orientation fixed with respect to the external magnetic field, the only rotation that is performed of the crystal with respect to the incoming neutron beam is a rotation of the -axis of the diffractometer, that is the vertical direction in the laboratory frame.The data collections are summarized in Table S9.
Regions of interest (ROIs) containing potential peaks were extracted from the raw images using a suite of in-house data reduction programs from the LLB.Based on these ROIs, the crystal orientations were refined by fitting an orientation matrix with the 100 K structure of 1 and the 150 K structure of 2. Only ROIs, for which the deviation of either h, k, or l was less than 0.2 from the closest integer value, were kept and used to calculate flipping ratios for the further analysis.The flipping ratios obtained in this way are plotted as a function of 2 in Figure S9 and Figure S10 for 1 and 2 respectively.
Table S9.Summary of data collections and the reduced data from 1 and 2. Numbers in parentheses constitute a second setting of the detector.Notice that for 1, the first orientation was measured by making slices of reciprocal space around expected peak positions, and therefore, the minimum, maximum and incremental values of phi for a single rotation are not applicable here and therefore not noted in the table.An important success criterium of the PND experiment is the directions of the magnetic field with respect to the crystal.Preferably, these directions should be different enough to probe different directions of the molecular magnetization, for the refined parameters to be uncorrelated.To judge whether this has been achieved, we have mapped the magnetic field directions based on the orientation matrices for each orientation onto the molecule in the asymmetric unit of the crystal structure in Figure S11.As seen in Figure S11, a large spread in the field directions was successfully achieved for 1.For 2, we observe that one set of field directions were in the horizontal plane (blue, orange), while the other two field directions were both along the axial direction and transverse directions (red, green) of the symmetry related molecules in the unit cell (Figure S12).Given the likeness of orientations 1 and 2 and of orientations 3 and 4, effectively the (red,green)orientations are probing both the axial and transverse elements, whereas the (blue,orange)orientations are only probing transverse elements of the magnetic susceptibility tensor for 2.
The refinement of the atomic site susceptibility tensor for 1 is straightforward, due to the wellseparated field orientations, the overall number of flipping ratios, and their relatively small standard deviations.The final model is shown in the main text, and the agreement between the data and the final model is shown in Figure S13 and Table S10.For 2 on the other hand, due to the scarcity of well-determined data, we approach the fit to these data with more caution.The two orientations for 2 with a "equatorial" magnetic field (3 and 4)   contain very few flipping ratios, but we notice that the flipping ratios are all close to 1, and that the flipping ratios for the orientations with the magnetic field in an "axial" direction to a larger extent show a deviation from 1.This follows the expected pattern, assuming that the primary anisotropy axis is along the O-Ln-O-direction, because magnetic fields that are perpendicular to the easy axis direction in an anisotropic compound will only induce small magnetic moments, giving rise to small deviations of the flipping ratio from 1.We note that a similar pattern is observed for 1, where the first orientation is almost perfectly aligned with the horizontal direction within the molecule, and the flipping ratios here are also very close to 1.These flipping ratios are all important in the refinement of the atomic susceptibility tensors, as the measurements that show the near absence of a magnetic moment, when the magnetic field is applied along these directions impose restraints on the maximum values of the atomic susceptibility tensor of 2.
As an exemplification of this happening, we attempt a refinement of the susceptibility tensor for 2 in the case where orientation 3 and 4 have been omitted from the refinement (model 1).In this model, the main anisotropy axis is oriented towards the horizontal plane within the molecule.Furthermore, the eigenvalues, that are in this case unbounded by the flipping ratios close to 1 in the transverse plane, increase dramatically, and the susceptibility tensor in that case refines to give eigenvalues of 47    −1 , 7    −1 and 2    −1 .The agreement factors between this model and the data are shown in Table S11.They are not markedly different from the agreement factors for some of the other models that we have tried with this data (vide infra), but we note that the susceptibility value of 47    −1 is unrealistically large for a single Ho(III) ion.The susceptibility tensor refined for this model is visualized on top of the molecular structure of 2 in Figure S14.Having tested the effect of the omission of orientations 3 and 4, and the exclusion of 4 reflections from orientation 2 on the refinement of the susceptibility tensor for 2, we also decided to test the robustness of the refinement.To do this, instead of refining the susceptibility tensor from different combinations of data, we assume an atomic susceptibility tensor for Ho(III) in 2 and then calculate the agreement with the data.This is done by deciding on the direction of the magnetic easy-axis and then building a set of orthogonal eigenvectors from this direction.Using the matrix diagonalization relation where  is the matrix of eigenvectors,  is a diagonal matrix containing the eigenvalues, and  is the   S11.

Fig. S1
Fig. S1The powder X-ray diffraction pattern of 2. The black line represents the simulated powder Xray diffraction pattern generated from single-crystal data collected at 150 K, and the red line represents the experimental data measured at ambient temperature.

Fig. S6 χ″Μ
Fig.S6χ″Μ vs χ′Μ plot of the AC magnetic susceptibility of 2 in zero dc field.The solid lines correspond to the best fit to Debye's law.7

Fig. S8
Fig. S8 Ab initio computed gzz orientation of the ground pseudo doublets for complex 2. Colour code: Ho, magenta; O, red; N, blue; P, pink; C, gray; I, dark yellow.Hydrogens atoms, co-crystalized HMPA molecules and disorder components are omitted for clarity.

Figure S9 .
Figure S9.Flipping ratios for 1 plotted as a function of scattering angle, 2, for orientations 1, 2 and 3 respectively.The blue line at a flipping ratio of 1 indicates the expected value in the absence of a magnetic structure.

Figure S10 .
Figure S10.Flipping ratios for 2 plotted as a function of scattering angle, 2 for orientations 1, 2, 3 and 4 respectively.The blue line at a flipping ratio of 1 indicates the expected value in the absence of a magnetic structure.The flipping ratios marked with a red circle for orientation 2 were omitted for some of the models attempted for 2.

Figure S11 .
Figure S11.Molecular structures of 1 (left) and 2 (right) shown as ball-and-stick models and with hydrogen atoms omitted.Magnetic field for orientations 1, 2, 3 and 4 are shown as red, green, blue, and orange arrows, respectively.Ln: teal, O: red, P: orange, N: blue, C: grey.

Figure S12 .
Figure S12.Unit cell of 2 with all 4 symmetry-related molecules and their relation to the magnetic field directions of the 4 crystal orientations described in the text.Hydrogen atoms, HMPA cocrystallized molecules and counterions have been omitted.Roughly speaking, the magnetic fields of orientation 1 and 2 (red and green) are probing both the axial and transverse elements of the

Figure S13 .
Figure S13.Agreement between model and data for 1.The plots show the calculated and experimental flipping ratios for orientation 1, orientation 2, a zoom of orientation 2 and orientation 3 from top left to bottom right respectively.

Figure S14 .
Figure S14.Atomic susceptibility tensor of 2 refined without orientation 3 and 4 to show the influence of refining an atomic susceptibility tensor without the influence of the transverse orientations.
simulated magnetic susceptibility tensor, we can represent the magnetic susceptibility tensor by any set of eigenvalues along these axes.The axes used as eigenvectors of the magnetic susceptibility tensor for our simulations are shown on top of the molecular structure of 2 in Figure S15.

Figure S15 .
Figure S15.Axes used for the simulation of a susceptibility tensor to compare with the data for 2.

Figure S16 .
Figure S16.Agreement between model 3 and the data for 2. The plots show calculated vs. experimental flipping ratios for orientation 1, 2, 3, and 4 from top left to bottom right respectively.

Figure S17 .
Figure S17.Visualization of models tested for their agreement against the experimental data.Models 2, 3, 4, and 5 are described in the text and presented from left to right.

Table S5 .
The ab initio computed crystal field parameters for complex 2.

Table S6 .
15nsor elements (  ) in the basis of the Cambridge Crystallographic Subroutine Library,15total  2 -value, easy-axis discrepancy compared to the ND-direction, and similarity index, S12, for the refinement of PND-data for 3 against three different structural models for simulation of nuclear structure factors.Unit on   's is    −1 .

Table S7 .
15nsor elements (  ) in the basis of the Cambridge Crystallographic Subroutine Library,15total  2 -value, easy-axis discrepancy compared to the ND-direction, and similarity index, S12, for the refinement of PND-data for 4 against four different structural models for simulation of nuclear structure factors.Unit on   's is    −1 .

Table S8 .
R-values of the refinement against X-ray diffraction data with short (Hshort) and elongated (Hlong) bonds to H-atoms for 1 and 2.

Table S10 .
Agreement between model and data for 1 represented by  2 -values based on the flipping ratios.

Table S11 .
Agreement factors between the 4 orientations measured for 2 and the 5 models described in the text.